Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



The computation of resonances in open systems using a perfectly matched layer

Authors: Seungil Kim and Joseph E. Pasciak
Journal: Math. Comp. 78 (2009), 1375-1398
MSC (2000): Primary 65N30, 78M10
Published electronically: February 6, 2009
MathSciNet review: 2501055
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we consider the problem of computing resonances in open systems. We first characterize resonances in terms of (improper) eigenfunctions of the Helmholtz operator on an unbounded domain. The perfectly matched layer (PML) technique has been successfully applied to the computation of scattering problems. We shall see that the application of PML converts the resonance problem to a standard eigenvalue problem (still on an infinite domain). This new eigenvalue problem involves an operator which resembles the original Helmholtz equation transformed by a complex shift in the coordinate system. Our goal will be to approximate the shifted operator first by replacing the infinite domain by a finite (computational) domain with a convenient boundary condition and second by applying finite elements on the computational domain. We shall prove that the first of these steps leads to eigenvalue convergence (to the desired resonance values) which is free from spurious computational eigenvalues provided that the size of computational domain is sufficiently large. The analysis of the second step is classical. Finally, we illustrate the behavior of the method applied to numerical experiments in one and two spatial dimensions.

References [Enhancements On Off] (What's this?)

  • 1. J. Aguilar and J. M. Combes.
    A class of analytic perturbations for one-body Schrödinger Hamiltonian.
    Commun. Math. Phys., 22:269-279, 1971. MR 0345551 (49:10287)
  • 2. S. Balay, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang.
    PETSc users manual.
    Technical Report ANL-95/11 - Revision 2.1.5, Argonne National Laboratory, 2004.
  • 3. E. Balslev and J. M. Combes.
    Spectral properties of many body Schrödinger operators with dilation analytic interactions.
    Commun. Math. Phys., 22:280-294, 1971. MR 0345552 (49:10288)
  • 4. J.-P. Bérenger.
    A perfectly matched layer for the absorption of electromagnetic waves.
    J. Comput. Phys., 114(2):185-200, 1994. MR 1294924 (95e:78002)
  • 5. J.-P. Bérenger.
    Three-dimensional perfectly matched layer for the absorption of electromagnetic waves.
    J. Comput. Phys., 127(2):363-379, 1996. MR 1412240 (97h:78001)
  • 6. D. Boffi.
    Fortin operator and discrete compactness for edge elements.
    Numer. Math., 87(2):229-246, 2000. MR 1804657 (2001k:65168)
  • 7. D. Boffi, P. Fernandes, L. Gastaldi, and I. Perugia.
    Computational models of electromagnetic resonators: analysis of edge element approximation.
    SIAM J. Numer. Anal., 36(4):1264-1290 (electronic), 1999. MR 1701792 (2000g:65112)
  • 8. J. H. Bramble and J. E. Osborn.
    Rate of convergence estimates for nonselfadjoint eigenvalue approximations.
    Math. Comp., 27:525-549, 1973. MR 0366029 (51:2280)
  • 9. J. H. Bramble and J. E. Pasciak.
    Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems.
    Math. Comp., 76:597-614, 2007. MR 2291829 (2008b:65130)
  • 10. W. Chew and W. Weedon.
    A 3D perfectly matched medium for modified Maxwell's equations with stretched coordinates.
    Microwave Opt. Techno. Lett., 13(7):599-604, 1994.
  • 11. F. Collino and P. Monk.
    The perfectly matched layer in curvilinear coordinates.
    SIAM J. Sci. Comput., 19(6):2061-2090, 1998. MR 1638033 (99e:78029)
  • 12. D. Colton and R. Kress.
    Inverse Acoustic and Electromagnetic Scattering Theory.
    Springer-Verlag, New York, 1998. MR 1635980 (99c:35181)
  • 13. S. Hein, T. Hohage, and W. Koch.
    On resonances in open systems.
    J. Fluid Mech., 506:225-284, 2004. MR 2259489 (2007d:76204)
  • 14. S. Hein, T. Hohage, W. Koch, and J. Schöberl.
    Acoustic resonances in a high lift configuration.
    J. Fluid Mech., 582:179-202, 2007. MR 2331498
  • 15. V. Hernandez, J. E. Roman, and V. Vidal.
    SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems.
    ACM Transactions on Mathematical Software, 31(3):351-362, Sept. 2005. MR 2266798
  • 16. P. D. Hislop and I. M. Sigal.
    Introduction to Spectral Theory with Applications to Schrödinger Operators.
    Springer, Berlin-Heidelberg-New York, 1996. MR 1361167 (98h:47003)
  • 17. T. Kato.
    Perturbation Theory for Linear Operators.
    Springer-Verlag, Berlin-Heidelberg-New York, 1976. MR 0407617 (53:11389)
  • 18. S. Kim and J. E. Pasciak.
    Analysis of a Cartesian PML approximation to acoustic scattering problems in $ \mathbb{R}^2$.
    In preparation.
  • 19. F. Kukuchi.
    On a discrete compactness property for the Nédélec finite elements.
    J. Fac. Sci. Univ. Tokyo, Sect. 1A, Math, 36:479-490, 1989. MR 1039483 (91h:65173)
  • 20. M. Lassas and E. Somersalo.
    Analysis of the PML equations in general convex geometry.
    Proc. Roy. Soc. Edinburgh Sect. A, 131(5):1183-1207, 2001. MR 1862449 (2002k:35020)
  • 21. P. Monk.
    Finite Element Methods for Maxwell's Equations.
    Numerical Mathematics and Scientific Computation. Oxford Univeristy Press, Oxford, UK, 2003. MR 2059447 (2005d:65003)
  • 22. P. Monk and L. Demkowicz.
    Discrete compactness and the approximation of Maxwell's equations in $ {\mathbb{R}}\sp 3$.
    Math. Comp., 70(234):507-523, 2001. MR 1709155 (2001g:65156)
  • 23. J. E. Osborn.
    A note on a perturbation theorem for the matrix eigenvalue problem.
    Numer. Math., 13:152-153, 1969. MR 0246502 (39:7806)
  • 24. B. Simon.
    The theory of resonances for dilation analytic potentials and the foundations of time dependent perturbation theory.
    Ann. Math., 97:247-274, 1973. MR 0353896 (50:6378)
  • 25. L. N. Trefethen.
    Pseudospectra of linear operators.
    SIAM Rev., 39(3):383-406, 1997. MR 1469941 (98i:47004)
  • 26. M. Zworski.
    Numerical linear algebra and solvability of partial differential equations.
    Commun. Math. Phys., 229:293-307, 2002. MR 1923176 (2003i:35008)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 78M10

Retrieve articles in all journals with MSC (2000): 65N30, 78M10

Additional Information

Seungil Kim
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Joseph E. Pasciak
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Keywords: Perfectly matched layer, PML, resonances, nonsymmetric eigenvalue problem
Received by editor(s): July 9, 2007
Received by editor(s) in revised form: July 22, 2008
Published electronically: February 6, 2009
Additional Notes: This work was supported in part by the National Science Foundation through grant DMS-0609544.
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society